Securitization, Shadow Banking, and
Bank Regulationú
Julian Kolm
†
September, 2015
This paper studies the capital regulation of banks that choose whetherto become traditional, deposit taking banks or shadow banks that providecredit intermediation through securitization. If capital regulation only coverstraditional banks, it will lead to the emergence of excessively risky shadowbanks that can fully crowd out traditional banking. Optimal capital regulationincludes shadow banks and can prevent excessive risk taking and crowding-outby ensuring that banks’ equity per unit of investment remains the same forall securitization strategies. Capital requirements cause underinvestment thatcannot be mitigated by securitization.
JEL classification: G18; G21; G28; G32.Keywords: Securitization; Shadow Banks; Bank Riskiness; Bank Regulation;
Capital Regulation.
úI want to thank Gyöngyi Lóránth, Jean-Charles Rochet, Kose John, Christian Laux, Maarten Janssen,Katarzyna Rymuza, Anna Orthofer, Douglas Gale, and seminar and conference participants at theEuropean Economic Association congress 2013, HU Berlin, Austrian National Bank, and Universityof Vienna for helpful comments and discussion. I gratefully acknowledge financial support by theAustrian Science Fund for work on an earlier version of this paper at the University of Vienna.
†WU Vienna University of Economics and Business, Welthandelsplatz 1, 1020 Vienna, Austria. P:+43-1-31336-6340 E: [email protected]
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1. Introduction
This paper explores the regulation of shadow banks and traditional banks that engage in
securitization. Shadow banking describes a system of financial institutions that provide
credit intermediation parallel to traditional banks. To raise outside financing, shadow
banks rely on securitization, a process that transforms loan portfolios into marketable
securities that can be sold to investors such as mutual funds and insurance companies.
Unlike traditional banks, shadow banks do not issue deposits and as a result are typically
not subject to capital regulation. Securitization is also heavily used by traditional banks.
This allows banks to reduce their balance sheet size and capital requirements because
the securities they sell to investors do not appear on banks’ balance sheets for regulatory
purposes.
Securitization and shadow banks play central roles in the financial systems of the US
and Europe. The total amount of outstanding asset backed securities (ABS), including
mortgage backed securities, stood at about 10 trillion USD in the US and 1.9 trillion USD
in Europe by the end of 2014 (SIFMA, 2015a,b,c). Lending by shadow banks in 2013
amounted to over 50% of total lending by banks and shadow banks in the US and over 25%
in the Euro area (Valckx et al., 2014, p. 67). During the financial crisis of 2007-2009, the
market for asset backed commercial papers (ABCP) contracted significantly which further
led to distress in interbank markets (Acharya and Merrouche, 2012; Krishnamurthy et al.,
2014). In September 2008, the US government deemed it necessary to guarantee the
holdings in money market mutual funds (MMFs) to stop ensuing runs (Treasury, 2008;
Kacperczyk and Schnabl, 2013). Consequently, shadow banks and securitization have
become central topics of the international regulatory agenda (e.g., BCBS, 2014; FSB,
2014).
I study the optimal capital regulation of both traditional and shadow banks. The
paper’s main novelty is that banks can choose whether to become traditional banks
2
or shadow banks that provide parallel credit intermediation. The central goal of bank
regulation in this paper is to reduce banks’ risk taking.
I show that when banks’ risk taking opportunities are su�ciently large, capital regula-
tion cannot prevent excessive risk taking if it only constrains the behavior of traditional,
deposit taking banks and does not regulate shadow banks. When capital requirements
are such that traditional banks do not have an incentive to engage in excessive risk
taking, then some banks will find it profitable to become unregulated shadow banks,
which allows them to raise less costly equity. These shadow banks will engage in excessive
risk taking due to their high leverage.
If capital regulation is set without taking the possibility of shadow banking into
account, then all banks find it profitable to become shadow banks, fully crowding out
the traditional banks. These shadow banks will separate themselves into a group of
risky banks and a group of sound banks who signal their respective risk choice by using
di�erent securitization strategies. Some unregulated shadow banks may invest in negative
net present value projects.
To prevent excessive risk taking optimal capital regulation has to ensure that both
traditional and shadow banks must raise the same amount of equity per unit of investment
for all securitization strategies pursued in equilibrium. Hence, under optimal capital
regulation securitization cannot decrease the amount of costly equity that banks have
to raise, which prevents banks from increasing their investment. Banks are indi�erent
between becoming shadow banks or traditional banks. The optimal capital requirements
implicitly define limits on the amounts of securitization that traditional and shadow
banks can engage in, respectively. Regulators can use direct constraints on securitization
that limit the amount of leverage that can be obtained from securitization as substitutes
for capital regulation.
In an extension, I consider shadow banks that face lower bankruptcy costs than
3
traditional banks. In this case optimal capital requirements that deter risk taking by
shadow banks must be so high that shadow banking becomes unprofitable.
I derive these results in a model where banks have an incentive to pursue excessively
risky strategies because of high leverage and limited liability. Banks can choose the
riskiness of their investment portfolio, which remains their private information. The
expected return on investment di�ers among banks and also remains their private
information. Market participants and regulators, however, are aware of banks’ risk taking
opportunities and di�erences in expected returns. Banks can use three sources of funds:
publicly insured deposits, equity capital, and securitization. Traditional banks always
finance themselves with deposits and equity and may also issue securities. The presence
of a risk insensitive deposit insurance subsidizes risk taking by traditional banks. Shadow
banks di�er from traditional banks by not issuing deposits and financing themselves with
equity and securitization only.1
Equity capital is costly for banks, which makes it optimal for banks to minimize their
equity financing. Banks cannot sell their entire investment to outside investors because
of adverse selection. Banks can, however, raise large amounts of outside finance by
issuing securities. To reduce adverse selection banks collateralize theses securities with
the proceeds from loan portfolios that are held by distinct legal entities know as special
purpose vehicles (SPVs). These SPVs also typically receive liquidity insurance by their
sponsoring institutions that ensure that senior tranches constitute senior claims on the
sponsor (Acharya et al., 2013). By retaining the risk associated with junior tranches
and liquidity insurance, banks can credibly signal the profitability of their investment,
overcoming the adverse selection problem. When banks sells securities through SPVs
outside investors claims do not appear on banks’ balance sheets which reduces balance
1As described in Pozsar et al. (2012), the credit intermediation by the shadow banking system isoften spread out over several institutions that form a financing chain between ultimate lenders andborrowers. For simplicity, the model abstracts away this institutional feature and treats shadowbanks as single institutions that provide credit intermediation.
4
sheet size relative to the amount of investment. Selling securities increases banks’ risk
taking incentives because it increases their leverage while keeping them exposed to their
investment risk, which increases the option value of bankruptcy. These risk taking
incentives are not present in models where banks sell their entire loan portfolio because
they are not exposed to the loans’ riskiness once sold.
Capital regulation can provide incentives against risk taking by reducing banks’ leverage.
Because banks’ riskiness is not observable, regulators cannot rely on risk weighted capital
requirements. Capital regulation can, however, depend on the extent of securitization in
which banks engage. Basel III, for example, imposes higher capital requirements for banks’
securitization activities (BCBS, 2014). To ensure that traditional banks do not engage
in excessive risk taking, capital requirements must ensure that their equity does not fall
below a certain threshold. This ensures that less profitable banks, which would otherwise
engage in excessive risk taking, will not find it profitable to become traditional banks
due to their cost of capital. The necessary capital requirements increase in bank risk
taking opportunities that increase the option value of bankruptcy. Because securitization
reduces banks’ balance sheet size, on-balance-sheet capital requirements must increase
with higher levels of securitization. This implies that capital requirements must prevent
deposit-issuing traditional banks from engaging in very high levels of securitization, for
which the o� balance sheet leverage would always cause traditional banks to engage in
excessive risk taking.
If shadow banks remain unregulated, they have the option to engage in high levels of
securitization which allows them to reduce the amount of costly equity they have to raise.
Becoming a shadow bank will be profitable when the capital requirements for traditional
banks are high because risk taking opportunities are high. Banks with low expected
returns will engage in excessive risk taking because of the high leverage they obtain as
shadow banks. Crowding-out of the traditional banking sector occurs because banks are
5
able to signal their risk taking behavior. Shadow banks signal low risk taking by choosing
a level of securitization that lies between those of traditional banks and shadow banks
that take excessive risk. More profitable banks, which do not engage in excessive risk
taking, can thus benefit from higher leverage without su�ering from adverse selection.
Overinvestment by shadow banks occurs because banks cannot perfectly signal their
type, which leads to the cross subsidization of banks with negative NPV projects by
other banks that engage in excessive risk taking.
Optimal capital requirements must ensure that the amount of capital that banks must
raise remains constant for all levels of securitization of shadow banks and traditional
banks. Otherwise some securitization strategies would result in lower capital costs and
higher risk taking incentives. This results in underinvestment irrespective of banks’
securitization strategies because equity is costly for banks. Banks are indi�erent between
becoming traditional and shadow banks because they face the same capital costs. Such
capital regulation must require shadow banks to hold additional safe and liquid assets
financed by equity on their balance sheet because otherwise their o� balance sheet
leverage achieved thorough high levels of securitization would result in excessive risk
taking.
Shadow banking is only profitable when it allows for high levels of securitization that
increase leverage. Hence, shadow banking can be prevented by constraining the types of
securities that can be sold. Such regulation can substitute for the capital regulation of
shadow banks.
If shadow banks face lower bankruptcy costs than traditional banks, then, in order to
prevent risk taking, the capital requirements for shadow banks must be higher. Higher
capital requirements makes shadow banking unprofitable because they increase capital
costs and banks that do not engage in risk taking do not benefit from lower bankruptcy
costs.
6
2. Related Literature
Pozsar et al. (2012) provide an excellent overview of the shadow banking system that also
describes how shadow banks provide credit in parallel to the traditional banking system.2
The securitization process is described in detail by Gorton and Metrick (2013). Several
studies have documented the use of securitization for regulatory arbitrage to decrease
regulatory capital requirements (Jones, 2000; Calomiris and Mason, 2004; Ambrose et al.,
2005; Acharya et al., 2013; Gorton and Metrick, 2010). The idea that securitization
can increase the riskiness of banks goes back at least to the eighties and papers such as
Greenbaum and Thakor (1987), James (1988), and Santomero and Trester (1998). On
the other hand it has been pointed out that securitization may also be used to achieve
better risk sharing and lower funding cost by Benveniste and Berger (1987), Allen and
Gale (1989), Gorton and Pennacchi (1995), and Chiesa (2008), for example. Among other
papers Fender and Mitchel (2009) have explored how di�erent securitization structures
interact with banks’ moral hazard problems.
The two most closely related papers in the literature are Cerasi and Rochet (2012) and
Plantin (2014). These papers explore the possibilities for banks to refinance themselves
through securitization activities. In both papers, securitization is valuable because it is the
only available source of finance when additional investment opportunities arise. In Plantin
(2014), banks can sell securities to shadow banks when other funding opportunities are
not available. Selling securities to shadow banks allows traditional banks to circumvent
leverage restrictions because securitization activities cannot be observed by the regulator.
Due to this assumption, the paper cannot discuss capital requirements for securitization
activities, which are central to the present paper. Cerasi and Rochet (2012) consider the
optimal capital regulation for securitization activities in a framework where equity capital
provides banks with monitoring incentives. They show that the amount of lending should
2Valckx et al. (2014, p. 69, fig.2.3) contains a helpful simple graphical representation.
7
be limited to a multiple of the banks’ equity irrespective of the banks’ securitization
decision. This multiple, however, depends on the macroeconomic conditions of the
economy. I obtain a similar result based on a di�erent modeling approach. In their model,
capital ensures that the banks’ stake in higher returns generated by monitoring are large
enough to cover the monitoring cost. In my model, capital ensures that the banks’ share
in losses is large enough such that gambling on the option value of bankruptcy is not
profitable.
The main di�erence between my paper and Cerasi and Rochet (2012) and Plantin
(2014) is that I explore both traditional banks’ refinancing from outside investors through
securitization and the possibility for banks to become shadow banks that provide credit
intermediation parallel to the traditional banking sector. Pozsar et al. (2012) calls these
two phenomena the internal and external shadow banking system, respectively. Hence
my model encompasses a larger set of shadow banking activities than Cerasi and Rochet
(2012) and Plantin (2014), which only consider the internal shadow banking system that
refinances traditional banks via securitization. I show that the regulation of the external
shadow banking system is necessary for e�ective capital regulation.
In macroeconomics, shadow banks that operate parallel to traditional banks appear
in Verona et al. (2013) and Mazelis (2015). These papers focus on monetary policy
transmission and do not study bank regulation.
Nicolò and Pelizzon (2008) consider a model where banks engage in securitization to
reduce their capital requirements. They do not discuss optimal capital regulation, but
instead assume that capital requirements are always such that banks’ capital fully covers
their maximum possible losses. Hence securitization only reduces banks’ capital when it
reduces their riskiness.
Restrictions on securitization activities have been discussed by Gorton and Metrick
(2010) and Kashyap et al. (2010) without providing a formal model. Current recommen-
8
dations by the Financial Stability Board (FSB, 2014) include such restrictions under the
title of product based regulation.
From a modeling perspective, the present paper is based on Freixas et al. (2007). They
determine optimal capital requirements for banking conglomerates that can engage in
excessive risk taking. I extend their model to discuss the regulation of securitization
activities3 and shadow banks.
3. Model
Banks have exclusive access to an investment opportunity with expected return R œ
[Rl, R
h]. Banks di�er in returns R, which determine their type. This captures a situation
where banks specialize in information gathering in certain market segments such as
specific industries. Banks can choose the riskiness of their investment by choosing
investments with certain return correlations. Investments generate a strictly positive
payo�
X(R, b) =
Y___]
___[
R + b, with probability 1/2
R ≠ b, with probability 1/2
where b œ [0, B] measures the bank’s risk taking. Investment costs are normalized to one,
and the risk free interest rate is normalized to zero. An investment opportunity has a
positive net present value (NPV) if E[X] = R Ø 1.4
Banks’ types are continuously distributed over the interval [Rl, Rh]. To ensure that
banks with positive and negative NPV projects will both be present in the model
Rh > 1 > Rl > B. The cumulative distribution function of banks’ types is denoted by
3They do consider loan sales, which they call securitization. Their model, however, is such that loansales to outside investors will never be optimal in equilibrium, which is similar to my result in Section4.3.
4The probabilities 1/2 are only chosen for the ease of exposition and are not necessary for the qualitativeresults.
9
F (R). I assume that E[R] is su�ciently low so that adverse selection prevents banks
from selling their entire investment to outside investors, which is discussed in Section 4.3.
Banks’ portfolios of financial assets are more opaque and can more easily be changed
than the investments of traditional firms (Flannery, 1994), which makes it di�cult for
outsiders to evaluate banks’ investments. To capture this opaqueness I assume that
banks’ investment characteristics (R, b) are private information.5
3.1. Traditional and Shadow Banks
Banks have access to three sources of finance: equity, publicly insured deposits, and
selling securities to outside investors. A bank’s financing decision determines whether it
is a traditional or a shadow bank. Traditional banks take deposits, issue equity and may
use securitization to finance themselves. Shadow banks exclusively rely on equity and
securitization.6 Shadow banks include institutions that do not have a bank license and
hence cannot issue deposits as well as banks that in principal could take deposits but
are prevented from doing so by capital requirements, which are formally introduced in
Section 3.2.
To engage in securitization banks create special purpose vehicles (SPVs) that issue
asset backed securities (ABS) to outside investors. The SPV and the ABS contracts are
structured such that outside investors (that buy senior tranches) obtain senior claims on
the bank’s entire investment portfolio. To achieve this, banks transfer assets as collateral
to their SPVs and provide credit and liquidity guarantees. The transfer of assets to the
SPV removes them from the bank’s balance sheet and ensure that they are no caught up
5This is not meant to claim that regulation that conditions on banks’ riskiness is not or should notbe important. It acknowledges, however, the fact that regulators have to account for unobservablecharacteristics on top of of some observable risk characteristics. The model abstracts from theobservable characteristics of banks’ portfolios for simplicity.
6Shadow banks di�er from other firms that rely on market finance by making financial investments.These investments, as already mentioned, are more opaque and allow for higher levels of privateinformation risk taking.
10
in possible bankruptcy proceedings. The value of the collateral is weakly higher than the
claims of outside investors at the time of issuance. Liquidity and credit guarantees are
structured such that they provide seniority by priority and ensure that investors are fully
paid even if the value of the collateral held by the SPV falls short of their claims. The
prevalence and structure of credit and liquidity guarantees is documented in Acharya
et al. (2013). Gorton and Souleles (2006) discuss the legal issues involved in ensuring
that the transfer of assets to the SPV is upheld in the case of bankruptcy.
Outside investors that buy senior ABS tranches with a face value of A thus obtain
claims that o�er cash flows min{A, X}. Banks make outside investors a take it or leave
it o�er to buy their SPVs’ senior tranches at a price P . Outside investors are assumed
to be risk neutral. Hence outside investors will buy securities that, given their beliefs,
provide an expected return larger or equal to zero. The bank retains the right to the
cash flow max{X ≠ A, 0} and receives the SPV’s proceeds from selling securities P . The
structure of the SPVs ensures that the claims of outside investors buying securities do
not appear on banks’ balance sheets for regulatory purposes (Acharya et al., 2013). Thus,
the size of banks’ balance sheets after selling securities is 1 ≠ P .
Banks also raise equity E and possibly deposits D. I abstract away from other forms of
on-balance-sheet outside finance such as bonds. The balance sheet identity implies that
D © 1 ≠ P ≠ E. Negative amounts of deposits D < 0 constitute holdings of additional
safe and liquid assets. I assume that funds raised from depositors or outside investors
cannot be used to pay out dividends before investment returns have realized.
The following Definition formalizes the distinction between traditional, deposit taking
banks and shadow banks that raise their entire funds from equity and securitization.
Definition 1. A bank is a shadow bank when P + E Ø 1 and it is a traditional bank
when P + E < 1.
Equity financing carries an opportunity cost for banks’ current shareholders that
11
forego other investment opportunities because they are not able to sell their bank entire
investment (which will be shown in Section 4.3). The opportunity cost of providing one
unit of equity is 1 + Ÿ. Another reason commonly used in the literature to motivate
higher costs of equity are tax disadvantages.7
Depositors are fully protected by a public deposit insurance, and hence, accept deposits
with an interest rate of zero. Since individual bank risk taking is not observable, it cannot
be reflected in a deposit insurance premium, which thus must be a flat fee. For the rest
of the analysis I set it to zero, which provides a subsidy for the potential risk taking of
traditional banks.
When a bank goes bankrupt shareholders incur private costs ’ that include losses of
franchise value and reputation as well as forgone investment opportunities. As discussed
above the claims of outside investors holding collateralized securities have a higher priority
in bankruptcy than depositors. Equity holders have the lowest priority and are protected
by limited liability.
3.2. Regulator
The regulator will set minimum capital requirements C(A) that determine a lower bound
for banks’ on-balance-sheet equity ratio
E
(1 ≠ P ) Ø C(A).
7Admati et al. (2011) have recently challenged the notion that bank capital is socially costly. Theirmain argument is based on Modigliani-Miller and they point out that several reasons for costlybank capital, such as taxes, are regulatory distortions. In my model, however, assets markets arecharacterized by asymmetric information and hence Modigliani-Miller will not hold. Moreover, thepresent paper’s main concern is regulation and the regulator’s objective will not directly include theprivate costs of equity. (see Definition 2 and Section 7). The private costs of equity will, however,concern the regulator via their impact on banks’ behavior. Recent empirical evidence shows that, inviolation of Modigliani-Miller, equity capital has some cost for banks (Miles et al., 2013; Baker andWurgler, 2015).
12
Since banks’ types R and their risk taking b are private information they cannot be
used to determine capital requirements. Capital requirements can be very sophisticated,
however, by taking into account a banks securitization choice A, which is an observable
characteristic of the contracts that are used for securitization.8
I assume that the regulator aims to ensure a sound banking system, and therefore sets
capital regulation such that all active banks choose to be sound. This is the primary
objective of international policy institutions such as the Basel Committee:
“The first Core Principle [of bank regulation] sets out the promotion of safety
and soundness of banks and the banking system as the primary objective for
banking supervision. Jurisdictions may assign other responsibilities to the
banking supervisor provided they do not conflict with this primary objective.”
(BCBS, 2012, p.4f)
The most standard motivation for this objective is to protect banks’ creditors or to
minimize the costs of public deposit insurance. But in this model, the outside investors
that finance shadow banks are risk neutral, sophisticated, and not publicly insured.
Shadow banks’ fragility nevertheless imposes externalities on other participants in the
financial system. Fire sales of assets caused by a bank’s bankruptcy can result in very
large falls in asset prices that also a�ect the holdings and solvency of other banks that are
protected by deposit insurance. A bank’s bankruptcy also a�ects its relationship borrowers
who cannot simply obtain comparable credit from other banks due to information
asymmetries.9 This externality equally a�ects shadow banks’ borrowers.
The fact that regulators consider the failure of banks costly even in the absence of
insured depositors was demonstrated when US regulators permitted Goldman Sachs and
Morgan Stanley to convert into bank holding companies, which gave them access to8Allowing capital requirements to depend in addition on the price P does not provide additional
insights.9A summary of the empirical evidence on the value of bank lending relationships can be found in
Degryse et al. (2009, p. 19�).
13
the public assistance set up to prevent the bankruptcy of commercial banks covered by
deposit insurance. The Financial Stability Board explains that
“[...] experience from the crisis demonstrates the capacity for some non-
bank entities and transactions to operate on a large scale in ways that create
bank-like risks to financial stability [...].” (FSB, 2014, p. ii)
Another recurring justification for the public support of banks during the recent financial
crises was to ensure the credit supply to firms.
Capital requirements increase banks funding costs because equity is costly. This reduces
banks profits and leads to underinvestment in positive NPV projects. Hence, among
those capital regulations that can ensure a sound banking system, the regulator has an
incentive to choose the capital regulation that minimizes underinvestment. Maximizing
banks’ profits minimizes this underinvestment.
Ranking capital regulations is not trivial because for many capital regulations there
exist multiple equilibria. The following definition of optimal capital regulation accounts
for the possibility of multiple equilibria by requiring that the optimal capital regulation
always maximizes the set of positive NPV projects that get financed in any equilibrium
where all banks are sound.
Definition 2. A capital regulation C(A) is optimal if
1. In all equilibria, all active banks choose to be sound.
2. For all other capital regulations, all equilibria where all banks are sound are such
that all banks obtain a weakly lower equilibrium payo� than in all equilibria under
C(A).
The very strong second requirement of this definition is appropriate because such a
capital regulation exists, as will be shown in Proposition 4. The social costs of bank
14
failures are explicitly modeled in Section 7, which connects optimal capital regulation in
the above sense with the maximization of the social surplus.
3.3. Outside Investors
When banks o�er securities to outside investors, they face an adverse selection problem
because their profitability R is private information. At the same time, the possibility
to engage in risk taking constitutes a moral hazard problem. The securities (A, P ) that
banks o�er constitute a two-dimensional signal, which outside investors use to form
beliefs about the banks’ profitability R and its choice of risk b. Outside investors will
buy any security that given their beliefs yields a non negative payo�.
I assume that the amount of equity raised by a bank E does not serve as a signal.
Outside investors, who observe E, ignore any deviation from the amount of equity that
maximizes a bank’s profits after selling securities subject to C(A). This assumption
reflects the di�culty of banks to commit to sustaining a certain equity position in the
future. In a more complete model, banks would have the option to reduce equity by
paying dividends after selling securities. In such a model, banks’ equity obviously cannot
serve as a credible signal.10
Outside investors are sophisticated and understand banks’ incentives and the signaling
value of securitization decisions. Consider a bank that deviates to a securitization strategy
o� the equilibrium path. Whether it could be more profitable to sell this security than
the equilibrium payo� depends on the bank’s type R. Outside investors anticipate for
which types of banks it would be profitable to successfully sell a certain security o� the
equilibrium path. Hence, outside investors believe that whenever a bank deviates, it
is of a type for which this deviation could be profitable.11 I also require that outside
10If E were to serve as a signal this would considerably complicate the analysis. I do not think thatit would yield additional insights because banks can signal their risk taking behavior (and to someextent their leverage) by choosing di�erent securitization strategies.
11When such a type does not exist, the beliefs of outside investors are not restricted.
15
CapitalrequirementsC(A) are set.
Banks observetheir own
profitability Rand decide tobe active.
Banks make investments, chooserisk b, raise equity and deposits,
and o↵er a security (A,P ).
Outsideinvestors buysecurities.
Outcomes ofrisky
investmentsand payo↵sare realized.
Signaling game
Figure 1: Time-line of Events
investors correctly anticipate a bank’s choice of b, which will be uniquely determined
given outside investors’ beliefs about the bank’s type, the securitization strategy (A, P ),
and capital regulation C(A). Such beliefs are a straightforward application of the logic
of the intuitive criterion (Cho and Kreps, 1987). While formally the intuitive criterion is
only defined for 2 player signaling games, I will simply refer to the intuitive criterion
when I apply the above restrictions to players’ beliefs.
Modeling securitization as a signaling game fits the OTC markets that are used to sell
securitization products. It di�ers from a setup where the price of a security is determined
by a zero profit constraint for outside investors because the corresponding price function
P (A) may not be well defined o� the equilibrium path. The reason is that for a given
A the sets of banks that are willing to sell such securities will depend on the price P .
This will be reflected in outside investors’ beliefs and hence there may not exist a price
at which outside investors are willing to buy and make zero profits.
3.4. Time-line
Figure 1 shows the time-line of events. Capital regulation is analyzed by studying how
capital requirements C(A) change the outcome of the subsequent signaling game between
banks and outside investors. Banks first observe the expected return of their investment
opportunity R. Banks then make their investments, choose their risk, raise equity and
16
deposits and o�er securities. For simplicity, I model these decisions as simultaneous.12
Outside investors buy securities depending on their beliefs. In the following analysis,
equilibrium refers to a perfect Bayesian equilibrium that satisfies the intuitive criterion.
The signaling game will typically have multiple equilibria.
4. Signaling Game
4.1. Banks’ Riskiness
The bank’s risk neutral shareholders make expected profits fi(R, E, A, P, b), to which I
refer simply as bank’s profits. A bank’s strategy (E, A, P, b) determines whether it can
default or not. A bank is called sound if it never defaults on its depositors and outside
investors always receive the face value A. This is the case when the return in low state
satisfies R ≠ b Ø A + (1 ≠ P ≠ E). The profits of a sound bank are
fiS(R, E, A, P, b) = R ≠ A ≠ (1 ≠ P ≠ E) ≠ E(1 + Ÿ).
A bank is called fragile if it goes bankrupt in the low return state. Bankruptcy occurs
in the low return state when R ≠ b Æ A + (1 ≠ P ≠ E), in which case equity holders
incur the private bankruptcy costs ’. The profits of a fragile bank, protected by limited
liability, are
fiF (R, E, A, P, b) = 1/2[R + b ≠ A ≠ (1 ≠ P ≠ E)] ≠ 1
/2’ ≠ E(1 + Ÿ).
Banks are active when their expected profit is larger than zero. Otherwise they are
inactive and create zero payo�. The threshold above which a bank makes positive profits
12In a more realistic model, banks would first raise equity and deposits, make some investments, securitizethem, and then use the proceeds to make further investments that will again be securitized, etc. Thetiming of the investment and financing decisions is not crucial for the results of the model.
17
is R(E, A, P ).
Banks’ profits fiS and fiF depend negatively on the amount of equity they raise. Since
equity does not serve as a signal, banks will choose E = (1 ≠ P )C(A). The profit of
a fragile bank depends positively on b. Hence, a bank that chooses to be fragile will
always choose b = B. A sound bank is indi�erent between all b for which it is sound. For
convenience, I assume that sound banks choose b = 0.
Whether a bank wants to become sound or fragile depends on the option value of
bankruptcy, which decreases with banks’ expected return ˆfiSˆR >
ˆfiFˆR > 0 and the amount
of equity financing ˆfiFˆE <
ˆfiSˆE < 0. Securitization makes risk taking more profitable. The
face value A increases the option value of bankruptcy ˆfiSˆA <
ˆfiFˆA < 0. Higher proceeds
from securitization P allow banks to reduce their equity for a fixed capital requirement.
When a bank prefers to be fragile it has the possibility to do so because
fiF (R, E, A, P, B) > fiS(R, E, A, P, 0) ∆ R ≠ B < A + (1 ≠ P ≠ E).
For a fixed equity and securitization strategy, we can thus define a threshold above which
banks will choose to be sound R(E, A, P ). A necessary but not su�cient condition for
fragile banks to exist in equilibrium is that the potential private benefits of risk taking
exceed the private bankruptcy costs (B ≠ ’) > 0. Positive bankruptcy costs ’ > 0 ensure
that banks do not always (weakly) prefer to maximize their risk taking in order to benefit
from limited liability. Throughout the paper, I assume (B ≠ ’) > 0 and ’ > 0; to simplify
notation drop b from the arguments of the profit functions fiS and fiF .
4.2. Signaling Value of Securitization
A bank’s securitization strategies only depends on whether the bank chooses to be sound
or fragile. Conditional on banks’ being sound or fragile, their respective preferences over
18
securitization strategies are the same. This can be seen from banks’ indi�erence curves
fiS(R, E, A, P ) Ø fiS(R, E
Õ, A
Õ, P
Õ) … (A ≠ A
Õ) ≠ (P ≠ P
Õ) Æ (E Õ ≠ E)Ÿ (1)
and
fiF (R, E, A, P ) Ø fiF (R, E
Õ, A
Õ, P
Õ) … (A ≠ A
Õ) ≠ (P ≠ P
Õ) Æ (E Õ ≠ E)(1 + 2Ÿ). (2)
These indi�erence curves do not directly depend on banks’ types R. A bank’s type only
determines whether the banks chooses to be sound sound or fragile. The securitization
preferences of fragile banks are aligned because all fragile banks go bankrupt with the
same probability.
In equilibrium, banks always choose an optimal securitization strategy. Hence (1)
implies that a sound bank is indi�erent between all securitization strategies that are used
by sound banks of di�erent types R in equilibrium. For fragile banks (2) has analogous
implications. Since sound banking becomes relatively more profitable than fragile banking
as a bank’s profitability R increases, there exists a threshold above which all banks prefer
to be sound R.
Lemma 1. Let S be the set of banks’ strategies that are chosen in equilibrium. S consists
of 2 components SS and SF such that
1. for all strategies (E, A, P ), (E Õ, A
Õ, P
Õ) œ SS the profits of sound banks satisfy
fiS(R, E, A, P, 0) = fiS(R, E
Õ, A
Õ, P
Õ, 0) for all R. (3)
2. for all strategies (E, A, P ), (E Õ, A
Õ, P
Õ) œ SF the profits of fragile banks satisfy
fiF (R, E, A, P ) = fiF (R, E
Õ, A
Õ, P
Õ) for all R. (4)
19
When SF is non empty then there exists a single threshold R(S) such that all banks with
R > R are sound banks that choose a strategy in SS and all active banks with R < R are
fragile banks that choose a strategy in SF .
This Lemma shows that banks cannot precisely signal their type. By choosing a
securitization strategy, a bank can at most signal whether its type is above or below the
threshold R and whether they are su�ciently profitable to become active.
Because banks’ preferences for di�erent securitization strategies only coarsely depend
on their types equilibrium refinements such as the intuitive criterion or D1 (Cho and
Kreps, 1987), which exploit di�erences in the bank types’ incentives to deviate, have only
limited power. Hence, these refinements cannot provide a unique equilibrium.
The intuitive criterion restricts outside investors beliefs such that banks are always able
to sell certain types of securities. Consider a securitization strategy o� the equilibrium
path (AÕ, P
Õ) that would be a profitable deviation for banks that prefer to be fragile.
Let R
F ((1 ≠ P
Õ)C(AÕ), A
Õ, P
Õ) denote the lowest type of bank that can make positive
profits when it chooses to securitize (AÕ, P
Õ) and be fragile. Banks are never active when
they expect to earn less than zero profits. Hence outside investors must believe that
the deviating bank’s profitability is larger or equal than R
F ((1 ≠ P
Õ)C(AÕ), A
Õ, P
Õ). This
implies that outside investors expect to make non-negative profits whenever
1/2(AÕ + R
F ((1 ≠ P
Õ)C(AÕ), A
Õ, P
Õ) ≠ B) ≠ P
Õ Ø 0. (5)
Consider now a di�erent deviation (AÕ, P
Õ) that would only be profitable for banks
that prefer to be sound. Outside investors will anticipate that the bank that o�ers the
security must be of a type that prefers to be sound. Because all sound banks always fully
pay back A, outside investors will always be willing to buy such securities when A
Õ Ø P
Õ.
Lemma 2. Assume the beliefs of outside investors satisfy the intuitive criterion. Let
20
(AÕ, P
Õ) be a securitization strategy o� the equilibrium path. Outside investors are always
willing to buy securities (AÕ, P
Õ) if this could be a profitable deviation for banks of which
(i) some prefer to be fragile and (5) holds or (ii) all prefer to be sound and A
Õ Ø P
Õ.
Note that su�ciently pessimistic beliefs about the types of banks that would sell
securities o� the equilibrium path can support equilibria where outside investors make
strictly positive profits. For example, outside investors make positive profits whenever a
bank with R > R
F sells a security that satisfies (5) on the equilibrium path.
4.3. Limits of Outside Financing
Banks never have a strict incentive to choose securitization strategies where P > 1. The
reason is that banks cannot pay out dividends before investment returns have realized
and thus, must always pay back any funds they receive and do not use for investment.
Formally, suppose a securitization strategy (A, P ) with P > 1 is sold in equilibrium.
Consider now the securitization strategy (AÕ, P
Õ) =1A ≠ (P ≠ 1), 1
2. From inspection of
the profit functions it follows that this securitization strategy is payo� equivalent for all
players. Thus, without loss of generality, I only consider P Æ 1.
Alternatively a banks’ initial shareholders could try to sell their bank’s entire investment
to outside investors. If a bank truly sells its entire investment, the initial shareholders
will not be a�ected by any bankruptcy costs that the new owners of the investment
might experience at a later point in time. Equity holders will only be willing to sell their
bank’s investment if they can recoup their initial investment outlay and make profits
P ≠ 1 > 0. This condition, however, is independent of the bank’s expected return R.
Hence if shareholders can sell their banks’ investments at a price P > 0 this will be
profitable for all banks that would not make positive profits otherwise. The resulting
adverse section prevents shareholders from selling their banks’ entire investments because
the average rate of return of banks’ investment opportunities is su�ciently negative.
21
Initial shareholders must thus always retain stakes in their bank’s investment.
Lemma 3. If E[R] < 1 ≠ B, a bank’s initial shareholders never sell their bank’s entire
investment in equilibrium.
Note that from a regulatory perspective, selling the entire investment (after choosing
its riskiness) would eliminate excessive risk taking. The reason is that initial shareholders’
profits would not depend on the bank’s risk when it is sold and hence, there are no
risk taking incentives. The adverse selection problem faced by banks thus results in
securitization structures that increase banks’ risk taking incentives.
5. Unregulated Shadow Banks
Before analyzing the optimal capital regulation I consider capital requirements that only
concern traditional banks, which resembles typical, real world bank regulations. Capital
regulation only concerns traditional banks when C(A) Æ 1 for all A since banks withE
1≠P > 1 are shadow banks by definition. Conditional on a bank being a shadow bank,
such a regulation does not impose any constraints except preventing them from issuing
deposits.
Capital regulation that does not constrain the behavior of shadow banks cannot prevent
excessive risk taking when banks’ risk shifting incentives are large. Banks’ risk shifting
incentives depend on the option value of bankruptcy that increases in banks’ risks taking
opportunities B and decreases in their bankruptcy costs ’.
Unregulated shadow banking allows banks bank with relatively low expected returns R
to make positive profits. These banks can make positive profits because the can securitize
large parts of their balance sheet which increases their leverage and reduces their equity
costs. Consider a bank with R = 1 + 12’. This bank breaks even when it sells securities
(AÕ, P
Õ) = (1 + B ≠ 12’, 1) for all C(AÕ) Æ 1. It is easy to check that this securitization
22
strategy satisfies (5) and thus Lemma 2 implies that outside investors are always willing
to buy securities (AÕ, P
Õ).13 When a bank with R > 1 + 12’ sells securities (AÕ
, P
Õ) it
su�ers from adverse section but still makes positive profits.
All banks with R Ø 1 + 12’ must thus be active in equilibrium, either as shadow banks
or traditional banks. The trade-o� between becoming a shadow bank or a traditional
bank depends on the relative amounts of equity that banks have to raise and the possible
costs of averse selection when selling securities to outside investors. Clearly, higher
capital requirements for traditional banks make becoming a shadow bank relatively more
profitable. Adverse selection increases in R and hence, banks with low R find shadow
banking relatively more profitable.
When capital requirements for traditional banks are high then banks with low R will
become shadow banks. These banks will engage in excessive risk taking due to their high
leverage, low R, and high B. Importantly this risk taking occurs although shadow banks
do not benefit from a deposit insurance subsidy and have to compensate outside investors
for the risk of a low return state. When capital requirements for traditional banks are
su�ciently low, then there are equilibria in which all active banks become traditional
banks. But to attain such equilibria capital requirements must be so low that banks with
small R will engage in excessive risk taking due to their high leverage and high B.
Proposition 1. If C(A) Æ 1 for all A and
B > ’
1 + 3Ÿ
2Ÿ
(6)
then there does not exist an equilibrium that satisfies the intuitive criterion where all
active banks are sound.
Proposition 1 implies that when banks have substantial private risk taking opportunities,13The securitization strategy (AÕ, P Õ) maximizes the profits of fragile banks with E = 1 subject to (5).
Hence 1 + 12 ’ is the lowest type of bank that will be active in all equilibria.
23
then any optimal capital regulation that prevents fragile banking must include shadow
banks. The on-balance-sheet capital requirements for shadow banks must exceed 100%
(C(A) > 1) to prevent fragility. To implement such capital requirements the regulator
can require shadow banks to hold additional safe and liquid assets (e.g., cash) financed
by equity on their balance sheets.14
5.1. Naive Capital Regulation
When capital regulation not only excludes shadow banks from regulation, but naively
ignores their existence for the purpose of bank regulation, shadow banks can fully crowd
out traditional banking. Crowding-out of the traditional banking sector is of course
problematic because banks provide essential services such as money-like claims and
payment systems. It is not clear to what extent less regulated shadow banks can take
over these functions.
Definition 3. A capital regulation is naive if (i) C(A) Æ 1 for all A and (ii) it would be
optimal if banks never chose to become shadow banks.
In general, there exist infinitely many naive capital regulations. I focus on the naive
capital regulation that maximizes the set of securitization levels A that can appear
in equilibrium. Any other naive capital regulation would arbitrarily reduce the set of
securitization strategies that traditional banks may use in equilibrium.
Lemma 4. The capital regulation
C
N(A) =
Y___]
___[
B≠’(1+Ÿ)(1≠A) A < 1 ≠ B≠’
1+Ÿ
1 otherwise
is naive.14Liquidity requirements such as those proposed by the Basel Committee BCBS (2013) require banks to
hold a certain amount of safe and liquid assets on their balance sheet.
24
For any other naive capital regulation C
Õ(A) and any A
Õ for which C
Õ(AÕ) ”= C
N(AÕ),
no traditional bank will use any securitization strategy (AÕ, P ) in any equilibrium under
C
Õ(A).
Capital regulation ensures that all active traditional banks are sound when all banks
that would prefer to be fragile prefer to remain inactive. This is possible because higher
capital requirements C make being fragile relatively less profitable, and thus the threshold
above which a bank wants to become sound R decreases. At the same time, it also
makes banking overall less profitable, which increases the threshold R above which a
bank wants to be active. The lowest capital requirement such that R Ø R for all (A, P )
and A < 1 ≠ B≠’1+Ÿ is given by C
N (A).15 At the same time, the capital regulation prevents
banks with securitization strategies where A Ø 1 ≠ B≠’1+Ÿ from becoming traditional banks
by requiring C
N(A) = 1, which implies that E = 1 ≠ P . Thus naive capital regulation
implicitly defines a limit on the amount of securitization that is admissible for traditional
banks. From the perspective of a naive regulator, who ignores the possibility of banks
becoming shadow banks, this constitutes a limit on the securitization activities for the
whole banking sector, ensuring a sound banking system.
The profits of sound banks are the same for di�erent securitization strategies with
A = P Æ 1≠ B≠’1+Ÿ . The reason is that banks’ equity costs (1≠P )CN (P )(1+Ÿ) = E(1+Ÿ)
remain constant. Thus sound banks are indi�erent between those securitization strategies
in equilibrium. When another naive capital regulation C
Õ(A) di�ers from C
N(A), two
cases can arise. First C
Õ(A) > C
N(A) for some A. This makes securitization strategies
with (A, P ) less profitable and hence they will not be used in equilibrium. Second,
C
Õ(AÕ) < C
N(AÕ) for some A
Õ. When this makes securitization strategies (AÕ, P ) more
profitable, some traditional banks will prefer to become fragile which violates the definition
of naive capital regulation.
15Freixas et al. (2007) already showed this for banks that do not engage in securitization.
25
5.2. Crowding-out of Traditional Banking
Under a naive capital regulation, two types of equilibria can emerge: either all banks
become sound traditional banks or all banks become shadow banks.
Proposition 2. Assume that the capital regulation is C
N(A). There does not exist a
banking system where both traditional and shadow banks are active.
In equilibrium fragile shadow banks are only active when they make strictly higher
profits than sound traditional banks. Otherwise they would not either not be active or
become sound banks traditional banks. Banks’ profits under under C
N are continuous
by construction. This implies that there exist securitization strategies o� the equilibrium
path that are only profitable for sound banks that become shadow banks. For su�ciently
small ‘, (A, P ) = (1≠ B≠’1+Ÿ +‘, 1≠ B≠’
1+Ÿ +‘) is such a strategy. Since these banks can signal
that they are sound Lemma 2 implies that they will not su�er from adverse selection
and outside investors are willing to by their securities.
From Proposition 1 it follows that fragile shadow banks will be active under naive
capital regulation when (6) holds. Following the above arguments this implies that sound
banks will become shadow banks as well and traditional banking ceases to exist.
Corollary 1. Assume that the capital regulation is C
N(A). If (6) holds, then all active
banks are shadow banks.
If shadow banks remain unregulated, the regulator could lower the capital requirements
of traditional banks to prevent crowding-out of traditional banks. But there exist multiple
equilibria with di�erent profits for fragile shadow banks. It follows that for a single
capital regulation there might exist di�erent equilibria where either full crowding-out
occurs or some traditional banks become fragile. This multiplicity of equilibria make it
impossible to implement a regulation that guarantees the existence of sound traditional
banks in the presence of unregulated shadow banks.16
16The extension in Section 9 shows that when the bankruptcy costs di�er between traditional and
26
5.3. Overinvestment by Shadow Banks
Another problem of naive capital regulation is that in some equilibria, banks with
negative net present value projects (R < 1) find it profitable to become active. Because
banks cannot single their precise type in equilibrium outside investors cannot distinguish
di�erent types of fragile banks. As a result, they may cross-subsidize less e�cient
fragile banks with the profits they make from more e�cient fragile banks.17 This cross
subsidization can make banks with negative NPV project profitable. Their investments
are ine�cient even without taking into account the private and social bankruptcy costs.
Proposition 3. Assume that the capital regulation is C
N and R ≥ U [Rl, Rh]. There
exists an equilibrium where banks with R ≠ 1 < 0 are active when
B >
2 + 3Ÿ
1 + 2Ÿ
’. (7)
6. Optimal Capital Regulation
The optimal capital regulation is such that whenever a bank prefers to be active it
prefers to be sound. This is the same mechanism that ensures that traditional banks are
always sound under C
N . Contrary to naive capital regulation, however optimal capital
regulation extends to shadow banks to ensure that for all securitization strategies (A, P )
an active bank will prefer to be sound.
Proposition 4. The capital regulation
C
ú(A) = B ≠ ’
(1 + Ÿ)(1 ≠ A)
is optimal.shadow banks, then naive capital regulation may lead to an equilibrium where fragile shadow banksand sound traditional banks coexist.
17This is a well understood mechanism in corporate finance (e.g., Tirole, 2006, sec. 6.2).
27
For any other optimal capital regulation C
Õ(A) and any A
Õ for which C
Õ(AÕ) ”= C
ú(AÕ),
no bank will use any securitization strategy (AÕ, P ) in any equilibrium under C
Õ(A).
Banks’ profits under any optimal capital regulation equal fiS(R, C
ú(0), 0, 0). Under
C
ú(A) banks are indi�erent between all securitization strategies where A = P < 1.
All active banks will prefer to be sound for all levels of securitization if for all A
R((1 ≠ P )Cú(A), A, P ) Ø R((1 ≠ P )Cú(A), A, P ). (8)
The optimal capital regulation is constructed such that (8) holds with equality for all
securitization strategies (A, A), and hence, holds for all securitization strategies with
A Ø P . As a result there does not exist a combination (E, A, P ) where all banks prefer
to be sound and sound banks make higher profits. This implies that C
ú is optimal and
that if another optimal capital regulation di�ers for some securitization level A, it must
be such that this securitization level is never used in equilibrium.
When banks engage in securitization the amount of equity they have to raise under
C
ú never decreases
E = (1 ≠ P )Cú(A) = 1 ≠ P
1 ≠ A
C
ú(0) Ø C
ú(0).
Banks profits under C
ú do not depend on the level of securitization they engage in
because they are always sound and hence can always sell securities with P = A, as shown
in Lemma 2. Banks are thus indi�erent between becoming shadow banks or traditional
banks because their profits are the same whether the choose a level of securitization
where C
ú(A) > 1 or not.
In practice, capital requirements such as C
ú(A) may appear to be excessive. The
reason is that banks have to satisfy high capital requirements for assets that are risk-free
in equilibrium, which will be reflected in historical returns. The low risk of the underlying
28
assets, however, is determined endogenously and does depend on the incentives provided
by the capital requirements. Optimal capital requirements must be excessive when banks’
risk taking is not observable and hence capital requirements cannot reflect banks’ risk
choice.
To implement the optimal capital regulation C
ú(A), regulators must have estimates
of the maximum amount of risk taking B, banks’ bankruptcy costs, and the cost of
equity. Importantly, regulators do not need to know an individual bank’s type R or
risk choice b, nor the distribution of banks’ types F (R). The optimal capital regulation
C
ú defines the maximum amount of securitization for traditional banks in the same
way as the naive capital regulation C
N . Shadow banks are required to hold additional
safe and liquid assets financed by equity on their balance sheets because the capital
regulation C
ú(A) exceeds 100% for A > 1 ≠ B≠’1+Ÿ . Note that C
ú(A) æ Œ as A æ 1. and
hence, the optimal capital regulation C
ú(A) implicitly defines a limit on the amount
of securitization that is admissible for banks. The reason is that when banks raise all
their financing from outside investors then the size of their balance sheet goes to zero
on-balance-sheet capital requirements loose their bite. Instead of preventing banks from
engaging in certain securitization activities those activities could in general be limited,
which is discussed in Section 8. Another alternative would be to consolidate banks’
securitization activities for regulatory purposes. When securitization activities are not
o�-balance-sheet, capital regulation directly constrains the banks overall leverage and
thus their risk taking incentives.
7. Welfare
Capital requirements for banks increase banks’ funding costs due to the cost of equity
Ÿ. This results in underinvestment because sound banks’ funding costs are larger than
the risk-free interest rate and hence banks with marginal positive NPV projects will not
29
become active. Since under optimal capital regulation banks’ profits do not depend on
their chosen level of securitization, Proposition 4 implies the following Corollary.
Corollary 2. Securitization activities cannot reduce the amount of underinvestment
under optimal capital regulation.
To connect the definition of optimal capital regulation with standard welfare analysis,
assume that a bank’s bankruptcy creates social costs „ which include the private costs ’.
Given the social costs of bankruptcy, a sound bank creates surplus R ≠ 1 and a fragile
bank creates a surplus R ≠ 1 ≠ 12„. Bank thus create a total surplus of
ˆ Rh
R
R ≠ 1 dF (R) ≠ˆ R
R
12„ dF (R).
A capital regulation that prevents excessive risk taking maximizes the total surplus if
the marginal surplus, which the least profitable active bank generates, is weakly lower
than the social cost of bankruptcy. In this case lower capital requirements cause marginal
banks that will be fragile to become active. The associated social costs of bankruptcy
would exceed the additional surplus .
Proposition 5. The optimal capital regulation maximizes the total surplus if the expected
social cost caused by bank fragility 12„ is higher than R(Cú(0), 0, 0) ≠ 1.
Otherwise it might increase welfare to allow for the presence of fragile banks in order to
generate additional investment. This will be the case if banks’ risk taking opportunities
B are very large compared to the social costs of bankruptcy. The reason is that for a
high B, capital requirements must be high to prevent excessive risk taking, which lead
to a high minimal expected return R for banks to become active.
The social costs of bankruptcy could be higher for traditional banks because public
deposit insurance has to bail out depositors if the bank fails. In this case, Proposition 5
remains valid when „ measures the social cost caused by the bankruptcy of a shadow
30
bank. When shadow banks’ social costs of bankruptcy do not satisfy the condition of
Proposition 5, it could be optimal to allow the presence of fragile shadow banks while
ensuring the soundness of traditional banks. This is problematic, however, because the
presence of shadow banks can crowd out traditional banks and lead to overinvestment,
as discussed in Sections 5.2 and 5.3.
8. Regulating Securitization
The optimal capital regulation implicitly constrains the amount of securitization tra-
ditional and shadow banks banks can engage in, respectively. Instead of regulating
securitization at the level of the banks, it is possible to restrict the types of securities
that can be traded in the market. Such restrictions limit the amount of funding and the
leverage that banks can obtain through securitization, which allows to prevent either
shadow banking, full crowding-out of traditional banking, or overinvestment by shadow
banks.
When capital regulation is naive, securitization restrictions can prevent shadow banking
by banning the use of all securitization activities that are only available to shadow banks
i.e., imposing A Æ 1 ≠ B≠’1+Ÿ . Banning all securitization strategies that would make it
profitable for sound banks to become shadow banks i.e., all securitization strategies
where fiS(1 ≠ P, A, P ) Ø fiS(CN(0), 0, 0), prevents crowding-out of traditional banking.
Restricting the admissible securitization activities to those where R((1≠P )C(A), A, P ) >
1 prevents overinvestment by shadow banks.
The literature typically conceives restrictions of tradable securities as minimum haircuts
(Gorton and Metrick, 2010; Kashyap et al., 2010), which restrict the amount of financing
for a given collateral value. Current recommendations by the Financial Stability Board
include both restrictions of securitization activities at the bank and the market level
(FSB, 2014).
31
It might be easier for regulators to implement restrictions on the types of securities that
market participants can use than to subject all the diverse institutions that constitute
the shadow banking sector to capital regulation. But unlike in the present model,
shadow banks and the securities they issue might serve some purpose that cannot be
fully substituted by other financial instruments. Capital regulation can ensure that
these activities will not lead to excessive risk taking while leaving their use to market
participants. This flexibility is lost by the direct regulation of securitization activities.
9. Di�erential Bankruptcy Costs.
Shadow banks are typically subject to fewer regulations than traditional banks that issue
insured deposits. Hence, it is often easier to create new shadow banks which increases
competition that decreases franchise values. Since the private bankruptcy costs ’ include
losses of franchise value this implies lower bankruptcy cost for shadow banks.
Assume that shadow banks face bankruptcy costs › that are lower than those of
traditional banks ’. The expected profit of a fragile bank is
fi
ÕF (R, E, A, P, b) = 1
/2[R + b ≠ A ≠ (1 ≠ P ≠ E)]
≠ 1/2[’1E+P <1 + ›(1 ≠ 1E+P <1)] ≠ E(1 + Ÿ)
where 1E+P <1 is an indicator function that takes the value of 1 if E + P < 1 (traditional
bank) and 0 if E + P Ø 1 (shadow bank).
Lower bankruptcy costs require higher capital requirements to prevent excessive risk
taking. Hence, the optimal capital requirements for shadow banks with lower bankruptcy
costs must be higher than for traditional banks. Sound shadow banks do not benefit from
lower bankruptcy cost, though and make strictly lower profits than sound traditional
banks that face lower optimal capital requirements. Hence in equilibrium all active banks
32
will be sound traditional banks. This shows that optimal capital regulation facing lower
bankruptcy costs of shadow banks must prevent the emergence of shadow banks.
Proposition 6. When › < ’, then any optimal capital regulation is such that in equilib-
rium no active bank will be a shadow bank.
Conversely, lower bankruptcy costs make fragile shadow banks more profitable when
shadow banking remains unregulated. Hence, capital requirements that do not constrain
the behavior of shadow banks lead to the emergence of fragile banks for a larger set of
parameters.
Proposition 7. If C(A) Æ 1 and the bankruptcy costs of shadow banks › > 0 are
su�ciently small, then there does not exist an equilibrium that satisfies the intuitive
criterion where all active banks are sound.
Di�erential bankruptcy costs, however, make it possible for traditional and fragile
banks to coexist in a segregating equilibrium under naive capital regulation. The presence
of fragile shadow banks under naive capital regulation does no longer necessitate full
crowding out of the traditional banking sector.
Proposition 8. Assume that the capital regulation is C
N(A) and let (AF, P
F ) = (1 +
B ≠ 12›, 1). There exists a segregating equilibrium where sound and traditional banks
coexist if
fi
ÕF (R, 1 ≠ P
F, A
F, P
F ) > fi
ÕF (R, C
N(0), 0, 0) (9)
and
fi
ÕF (R, 1 ≠ P
F, A
F, P
F ) < fi
ÕF (R, C
N(0), 1 ≠ C
N(0), 1 ≠ C
N(0)) (10)
With di�erential bankruptcy costs fragile shadow banks benefit from lower costs of
bankruptcy, unlike sound shadow banks that never go bankrupt. This di�erence makes
it possible that the following two conditions are met at the same time:
33
1. Fragile shadow banks find it profitable to be active i.e., (9) holds.
2. Any shadow banking activity that would be profitable for sound banks would also
be a profitable deviation for shadow banks i.e, (10) holds.18
As a result sound banks cannot signal their risk taking behavior when they become
shadow banks. Hence it is not profitable for sound banks to deviate from being a
traditional bank because they would have to pay an adverse selection premium if the
want to become shadow banks and sell the corresponding securities to outside investors.
10. Conclusions
Capital regulation can only prevent excessive risk taking when securitization activities and
shadow banks are properly taken into account. Risk taking can be prevented when the
amount of equity capital that banks have to raise per unit of investment does not decrease
with higher levels of securitization. Such capital requirements must require shadow banks
to hold additional capital that exceeds their on-balance-sheet investment. Regulating
securitization can substitute for capital regulation of shadow banks. Securitization
restrictions limit the amount of leverage that shadow banks can take, which has the same
e�ect as capital regulation.
If capital regulation only regulates the behavior of traditional, deposit taking banks,
then shadow banks will emerge and take excessive risk. In addition, unregulated shadow
banking might fully crowd out traditional banking, when capital regulation is only
concerned with ensuring that traditional deposit taking banks are sound. This potentially
jeopardizes additional functions of the traditional banking system such as the provision
of a payment system.
18Technically (9) and (10) imply that fiF (R, CN (0), 1 ≠ CN (0), 1 ≠ CN (0)) > fiF (R, CN (0), 0, 0). Dueto the construction of CN this holds if and only if › < ’.
34
Under optimal capital regulation, securitization cannot decrease banks’ capital costs
and hence, the underinvestment that is caused by capital regulation cannot be mitigated.
When shadow banks have lower bankruptcy costs than traditional banks, then to prevent
excessive risk taking their capital requirements per unit of investment must be higher
than for traditional banks. This makes sound shadow banking unprofitable. Hence,
shadow banks are likely not to be present in an optimally regulated banking system.
Overall the results in this paper indicate that in order to create a stable financial system,
it might be necessary to enact substantial changes to current regulation. While such
changes carry large uncertainties with them, the costs of not addressing the shortcomings
of current regulation can be huge, as experienced in the recent financial crises.
A. Proofs
A.1. Proof of Lemma 1.
Proof. Because equations 1 and 2 do not depend on R, equations 3 and 4 must hold for
all strategies that are used in equilibrium by sound and fragile banks, respectively.
Because ˆfiFˆR <
ˆfiSˆR , there exists a single type of bank R that would be indi�erent
between all equilibrium strategies S. All banks with R > R prefer to be sound banks
and choose a strategy in SS. All active banks with R < R prefer to be sound banks and
choose a strategy in SF .
A.2. Proof of Lemma 2.
Proof. Let D be the non empty set of banks that find a deviation (AÕ, P
Õ) profitable.
The intuitive criterion requires that when outside investors observe such a deviation,
they believe that the banks’ type is in D with probability 1.
In equilibrium all banks always make profits of at least zero. Hence, when some banks in
35
D prefer to be fragile then ˆfiFˆR <
ˆfiSˆR implies that the least profitable bank that could find
(AÕ, P
Õ) profitable is given by R
F ((1≠P
Õ)C(AÕ), A
Õ, P
Õ). Thus outside investors believe that
they make profits that are greater or equal to 1/2(AÕ +R
F ((1≠P
Õ)C(AÕ), A
Õ, P
Õ)≠B)≠P
Õ.
They will always buy securities (AÕ, P
Õ) if this implies non-negative profits.
When all banks in D prefer to be sound then outside investors anticipate that they
will earn profits greater or equal to A
Õ ≠ P
Õ which yields non-negative profits for outside
investors when P
Õ Æ A
Õ.
A.3. Proof of Lemma 3
Proof. Suppose some banks sell their entire investment in equilibrium for a price P
e.
Since the entire investment is sold, the shareholders of these banks will not be a�ected
by bankruptcy costs. Let G(R) denote the cumulative distribution of expected returns
of banks’ investments that are sold in equilibrium. Obviously, the price for selling the
entire investment must be unique and P
e> 1. Because it is impossible to pay dividends
before investment returns materialize and A Ø P for all securitization strategies, banks’
initial shareholders with R + B < 1 will make positive profits if and only if they sell their
banks’ investments. Because E[R] =´ Rh
Rl RdF (R) Æ 1 ≠ B it follows that the expected
profit from buying a bank’s investment
ˆ Rh
Rl
R ≠ P dG(R) ƈ Rh
Rl
R ≠ P dF (R) < 0
is negative. This cannot be an equilibrium because outside investors make negative
expected profits, which proves the Lemma.
36
A.4. Proof of Proposition 1
The proof uses the following Lemma 5, which describes a potential deviation that is
possible if the beliefs of outside investors satisfy the intuitive criterion.
Lemma 5. Assume that C(A) Æ 1. Every equilibrium where some banks’ profits are
lower or equal to fiF (R, 0, 1 + B ≠ 12’, 1) fails the intuitive criterion.
Proof. Suppose that in equilibrium some banks’ profits are lower or equal to fiF (R, 0, 1 +
B ≠ 12’, 1). Then the securitization strategy (1 + B ≠ 1
2’, 1) would be a profitable
deviation for these banks. Algebra yields that this securitization strategy satisfies 2 for
all C(A). Hence, banks will be able to sell these securities, which cannot be the case in
equilibrium
The Proof of Proposition 1 provides su�cient conditions under which no equilibrium
where all active banks are sound can satisfy the condition provided by Lemma 5.
Proof of Proposition 1. From Lemma 5 it follows that a bank can always make profits
fiF (R, 0, 1 + B ≠ 12’, 1) = 1
2(R ≠ 1 ≠ 12’). Hence all banks with R > 1 + 1
2’ must be active
in an equilibrium that satisfies the intuitive criterion.
If all banks with R > 1 + 12’ are active as sound banks then
fiS(1 + 12’, (1 ≠ P
Õ)C Õ(AÕ), A
Õ, P
Õ) Ø 0
for some securitization strategy (AÕ, P
Õ). This implies that
C
Õ(AÕ) Æ 2P
Õ + ’ ≠ 2A
Õ
2Ÿ(1 ≠ P
Õ) .
Some of these banks will not be sound, however, if
fiS(1 + 12’, (1 ≠ P
Õ)C Õ(AÕ), A
Õ, P
Õ) < fiF (1 + 12’, (1 ≠ P
Õ)C Õ(AÕ), A
Õ, P
Õ).
37
This condition implies that
P
Õ<
2A
Õ + 2A
ÕŸ + 2BŸ ≠ ’ ≠ 3Ÿ’
2 + 2Ÿ
.
Because in any equilibrium P Æ A must hold, the above condition is always satisfied
when2A
Õ + 2A
ÕŸ + 2BŸ ≠ ’ ≠ 3Ÿ’
2 + 2Ÿ
> A
Õ
which holds if and only if (6) holds.
A.5. Proof of Lemma 4
Proof. Suppose that banks never choose a securitization strategy with C(A) Ø 1. Thus
under C
N(A) banks will only choose securitization strategies with A < 1 ≠ B≠’1+Ÿ .
The capital regulation C
N is such that
fiS(R, (1 ≠ A)CN(A), A, A) = fiS(R, C
N(A), 0, 0)
for all A Æ 1 ≠ B≠’1+Ÿ . Hence banks must make profits that are fiS(R, C
N(A), A, A) in
equilibrium because they can always choose not to engage in securitization. Because
R((1 ≠ A)CN(A), A, A) = R((1 ≠ A)CN(A), A, A)
for all A < 1 ≠ B≠’1+Ÿ , C
N ensures that all active traditional banks will be sound.
Consider a di�erent naive capital regulation C
Õ(A). Suppose that C
Õ(AÕ) ”= C
N(AÕ)
for some A
Õ. Since in equilibrium outside investors demand P
Õ Æ A
Õ, fiS(R, (1 ≠
P
Õ)C Õ(AÕ), A
Õ, P
Õ) > fiS(R, C
N(0), 0, 0) implies that C
Õ(AÕ) < C
N(AÕ). But because
38
ˆfiFˆE <
ˆfiSˆE < 0 and ˆfiS
ˆP >
ˆfiFˆP > 0 this also implies that
R
F ((1 ≠ P
Õ)C Õ(AÕ), A
Õ, P
Õ) < R((1 ≠ P
Õ)C Õ(AÕ), A
Õ, P
Õ)
i.e., some active banks will be fragile. Hence if a securitization strategy (AÕ, P
Õ) is used in
equilibrium and C
Õ(AÕ) ”= C
N (AÕ), C
Õ(A) cannot be a naive capital regulation. Conversely
for any naive capital regulation C
Õ(A) and any securitization strategy (AÕ, P
Õ) that is
chosen by traditional banks in equilibrium C(AÕ) = C
N(AÕ).
A.6. Proof of Proposition 2
Proof. Consider a banking system with capital regulation C
N , where both traditional
and shadow banks coexist. C
N is chosen such that traditional banks either prefer to be
sound or to remain inactive. Hence traditional banks will be sound and choose some
securitization strategy A
S = P
S< 1 ≠ B≠’
1+Ÿ . From Proposition 1 it follows that under
C
N some fragile shadow banks will be active. Their profits must be strictly higher than
those of fragile traditional banks would be. Otherwise it would not be profitable for them
to remain active. Thus a fragile bank’s securitization strategy (AF, P
F ) must satisfy
fiF (R, (1 ≠ P
S)CN(A), A
S, P
S) < fiF (R, 1 ≠ P
F, A
F, P
F ). (11)
Sound banks would find any securitization strategy A = P > 1 ≠ B≠’1+Ÿ a profitable
deviation because it would reduce their capital requirements and associated costs per
unit of investment. From Lemma 2.2 it follows that sound banks could profitably deviate
when their deviation would not be profitable for fragile banks. Hence fragile banks’
profits must satisfy
fiF (R, 1 ≠ P
F, A
F, P
F ) Æ fiF (R,
B ≠ ’
1 + Ÿ
, 1 ≠ B ≠ ’
1 + Ÿ
, 1 ≠ B ≠ ’
1 + Ÿ
) (12)
39
to ensure that sound banks cannot profitably deviate.
The conditions (11) and (12) together imply
fiF (R, (1 ≠ P
S)CN(A), A
S, P
S) < fiF (R,
B ≠ ’
1 + k
, 1 ≠ B ≠ ’
1 + k
, 1 ≠ B ≠ ’
1 + k
).
Since C
N ensures that capital costs are the same for all A = P Æ 1 ≠ B≠’1+Ÿ this condition
implies a contradiction.
A.7. Proof of Proposition 3
Proof. Construct an equilibrium where all banks are shadow banks. Consider fragile
banks that choose (AF, P
F ) = (1 + B+2BŸ2+3Ÿ , 1) and sound banks that choose A
S = P
S =2(1+k)≠AF
1+2k .19 These securitization strategies are such that for R ≥ U [Rl, R
h] outside
investors make zero profits:
ˆ R
R((1≠P F )C(AF ),AF ,P F )1/2(R ≠ B + A
F ) ≠ P
FdF (R) = 0.
It is easy to check that R((1 ≠ P
F )CN(AF ), A
F, P
F ) < 1 when (7) holds.
Banks’ strategies constitute an equilibrium when outside investors beliefs are such that
only those deviations that satisfy the conditions of Lemma 2 are possible. Fragile banks
then do not have an incentive to deviate because fiF (R, 1 ≠ P
F, A
F, P
F ) is larger than
fiF (R, 0, 1 + B ≠ 12’, 1) which maximizes the profit of sound banks subject to condition
(5) in Lemma 2.
The strategy of sound banks satisfies fiF (R, 1, A
S, P
S) = fiF (R, 1, A
F, P
F ) and A
S =
P
S. Note that for E = (1≠P )CN the indi�erence curves of sound banks in the A-P -plane
are steeper than those of fragile banks but flatter than the 45¶ line. This implies that any
deviation with P
S Æ A
S that would be profitable for sound banks would also be profitable19This equilibrium is chosen such that it maximizes the profits of banks subject to the incentive
compatibility constraints that are discussed in the remainder of this proof.
40
for fragile banks. Outside investors will thus not be willing to buy any securities that
would be profitable deviations for sound banks.
In the above equilibrium, sound banks are of course only shadow banks when C
N (AS) =
1 … A
S Ø 1 ≠ B≠’1+Ÿ . Algebra yields that this is satisfied when (7) is satisfied.
A.8. Proof of Proposition 4
Proof. The capital regulation C
ú(A) is such that for all A < 1 the profits of banks satisfy
fiS(R, (1 ≠ A)Cú(A), A, A) = fiS(R, C
ú(A), 0, 0)
for all A. When A æ 1, C
ú(A) æ Œ and hence securitization strategies with A Ø 1
cannot be chosen by banks. Hence in equilibrium banks’ profits must equal fiS(R, (1 ≠
A)Cú(A), A, A). The capital regulation further implies that for all A < 1
R((1 ≠ A)Cú(A), A, A) = R((1 ≠ A)Cú(A), A, A).
It follows that all active banks must prefer to remain sound.
By the same arguments used in the proof of Lemma 4, there cannot exist another
optimal capital regulation C
Õ(A) where a securitization strategy (AÕ, P
Õ) is chosen in
equilibrium and C
Õ(AÕ) ”= C
ú(AÕ).
A.9. Proof of Proposition 5
Proof. Let (E, A, P ) be a securitization strategy that is chosen by sound banks in
equilibrium. From ˆfiFˆE <
ˆfiSˆE < 0 and ˆfiS
ˆP >
ˆfiFˆP > 0 it follows that fiS(E, A, P ) >
fiS((1 ≠ A)Cú(A), A, A) and P Æ A imply that R(E, A, P ) > R((1 ≠ A)Cú(A), A, A).
Fragile banks will only choose a di�erent securitization strategy from sound banks when
this is at least as profitable for them. Hence in any equilibrium S where fragile banks are
41
active R(S) > R((1 ≠ A)Cú(A), A, A).
The additional NPV that is created by allowing fragile banks is given by´ R(Cú(0),0,0)
R(S) R≠
1 dF (R) while the social costs of bankruptcy are given by´ R(S)
R(S)12x dF (R). Since R((1 ≠
A)Cú(A), A, A) = R((1 ≠ A)Cú(A), A, A) the social costs of bankruptcy exceed the
additional NPV when 12„ > (R(Cú(0), 0, 0) ≠ 1).
Hence the optimal capital regulation maximizes the total surplus.
A.10. Proof of Proposition 6
In what follows let A = 1 ≠ B≠’1+Ÿ ≠ ’≠›
1+2Ÿ .
Proof. Lower bankruptcy costs › < ’ make shadow banking more profitable. Banks
choose E = 1 ≠ P > (1 ≠ P )Cú(A) for C
ú(A) close to 1 because the increase in capital
cost from setting E = 1 ≠ P is outweighed by the decrease in bankruptcy costs for fragile
banks. It can easily be shown that
R((1 ≠ A)Cú(A), A, A) > R((1 ≠ A)Cú(A), A, A)
for A > A where fragile banks are shadow banks. Since any optimal capital regulation
must prevent risk taking, optimal capital requirements must be higher than C
ú(A) for
A > A.
For A Æ A and C
ú(A) < 0 the di�erence in bankruptcy costs is not su�cient to make
fragile shadow banking with E = 1 ≠ P > (1 ≠ P )Cú(A) profitable. Hence it follows from
the proof of Lemma 4 that under any optimal capital regulation sound banks’ profits
must equal fiS(R, C
ú(0), 0, 0). This implies that any optimal capital regulation must
equal C
ú(A) for some A Æ A.
For C(A) > C
ú(A) sound banks’ profits are strictly lower than fiS(R, C
ú(0), 0, 0).
Hence under optimal capital regulation sound banks will never choose securitization
42
strategies with A > A. Since C
ú(A) < 0 for A Æ A, banks under optimal capital
regulation will never be shadow banks.
A.11. Proof of Proposition 7
Proof. Analogous to the proof of Lemma 5 it can be shown that every equilibrium where
some banks’ profits are lower or equal to fi
ÕF (R, 0, 1+B ≠ 1
2›, 1) fails the intuitive criterion.
Analogous to the proof of Proposition 1 it can be shown that there does not exist an
equilibrium that satisfies the intuitive criterion where all active banks are sound if
B > ’ + ›
1 + Ÿ
2Ÿ
and C(A) Æ 1 for all A. Since B > ’ by assumption the proposition holds for su�ciently
small ›.
A.12. Proof of Proposition 8
Proof. Consider the equilibrium where all sound banks choose (AS, P
S) = (0, 0) and all
fragile banks choose (AF, P
F ). The beliefs of outside investors o� the equilibrium path
are such that Lemma 2 applies.
It is only profitable for shadow banks to sell securities (AF, P
F ) when (9) holds. There
is no profitable deviation for shadow banks because (1 + B ≠ 12›, 1) maximizes fragile
banks’ profits subject to (5).
Outside investors make zero profits from the least profitable active bank that sells
securities (AF, P
F ). Hence the expected profit of outside investors is positive because
R > R
F .
Sound banks would only be willing to deviate if they can make profits that are higher
43
than
fiS(R, C
N(0), 1 ≠ C
N(0), 1 ≠ C
N(0)) = fiS(R, C
N(0), 0, 0).
From equation (9) and the slopes of banks’ indi�erence curves in the A-P -plane for
E = (1 ≠ P )CN(A) it follows that any profitable deviation of sound banks would
be a profitable deviation for fragile banks as well. Since (AF, P
F ) maximizes sound
banks’ profits subject to (5), fiS(R, 1, A
F, P
F ) is the highest profit that sound banks
can make o� the equilibrium path. Simple algebra then shows that (5) implies that
fiS(R, C
N(0), 0, 0) > fiS(R, 1 ≠ P
F, A
F, P
F ) and hence there is no profitable deviation
for sound banks.
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